Cogito
Algebra 1 · Chapter 1 · Lesson 5
No Solution, or Every Solution
When the x disappears, read what is left.
11 problems · about 20 minutes · A-REI.B.3
What this lesson teaches
The student identifies equations with no solution and equations true for every value.
- If every x cancels, read the statement that remains.
- A false statement means no solution; a true one means every number works.
- Graphically these are parallel lines and one line drawn twice.
Warm Up
Straightforward practice. Get the method working first.
4 problems8x + 2 = 8x + 11. How many solutions?
Answer 0
Why None.
2x + 5 = 2x + 5. How many solutions? Use 99 for infinitely many.
Answer 99
Why The sides are identical.
7x + 1 = 7x + 4. How many solutions?
Answer 0
Why 1 = 4 is false.
4x = 4x. How many solutions? Use 99 for infinitely many.
Answer 99
Why Always true.
Build It Up
The same ideas with more to keep track of.
3 problems3x + 2 = 5x + 2. What is x?
Answer 0
Why 2x = 0.
Sort each result by what it means.
Answer Exactly one solution: x = 4 · No solution: 3 = 8 · Every number: 6 = 6
Why Ask whether the leftover statement is true, false, or an answer.
Two lines with the same slope and different intercepts. How many solutions?
Answer 0
Why Parallel.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problems2(x + 4) = 2x + 8. How many solutions? Use 99 for infinitely many.
Answer 99
Why Expand and compare.
6x + 3 = 6x − 3. How many solutions?
Answer 0
Why 3 = −3 is false.
The Contradiction: 3x + 2 = 3x + 8. How many solutions?
Answer 0
Why None.
The Identity: 5(x + 2) = 5x + 10. Is every number a solution? 1 for yes, 0 for no.
Answer 1
Why Yes — the two sides are identical.